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%% ladder.tex
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%% Made by Alex Nelson
%% Login   <alex@tomato>
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%% Started on  Wed Jun  3 16:59:03 2009 Alex Nelson
%% Last update Wed Jun  3 16:59:03 2009 Alex Nelson
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We see first of all that there is a state $|0\>$ which we will
call the ``\define{vacuum state}''. It is such that
\begin{equation}%\label{eq:}
a|0\>=0|0\>=0.
\end{equation}
That is, it is the lowest energy eigenstate. We find that in
explicit coordinates the function $f(Q)$ that does this is
\begin{subequations}
\begin{align}
\<Q|a|0\> &= a(f) \\
&= \left[Q+\frac{\partial}{\partial Q}\right]f(Q)\\
\Rightarrow -Qf(Q) &= f'(Q)\\
\Rightarrow -Q &= \frac{f'(Q)}{f(Q)}\\
\Rightarrow C_{1} - \frac{Q^{2}}{2} &= \ln(f(Q))\\
\Rightarrow f(Q) &= c\exp(-Q^{2}/2)
\end{align}
\end{subequations}
where $c$, $C_{1}$ are constants of integration. We find by
normalization
\begin{equation}%\label{eq:}
\<0|0\>=1\quad\iff\quad f(Q)= \<Q|0\> = \frac{1}{\sqrt[4]{\pi}}e^{-Q^{2}/2}.
\end{equation}
The energy of this eigenstate is then open to question. We would
\emph{like} it to be zero, as this is the vacuum, but we need to
\emph{prove} it.

First we should define the \define{Number Operator} as
\begin{equation}%\label{eq:}
N = a^{\dag}a.
\end{equation}
We see that
\begin{equation}%\label{eq:}
N|n\> = n|n\>
\end{equation}
and note the abuse of notation here: $n|n\>$ is the scalar $n$
multiplied with the ket $|n\>$. We chose this notation for the
ket because it gives us complete information about its number
eigenvalue (the scalar number $n$).

We can rewrite the Hamiltonian operator in terms of the number
operator
\begin{equation}%\label{eq:}
\widehat{H} = \hbar\omega\left(N+\frac{1}{2}\right).
\end{equation}
Observe then that the energy eigenvalue of the vacuum is
\begin{equation}%\label{eq:}
\widehat{H}|0\> = \frac{\hbar\omega}{2}|0\>.
\end{equation}
This is not really too good, since the \emph{vacuum has positive
  energy.} But what it also means is that we can write the energy
eigenvalues as
\begin{equation}%\label{eq:}
E_{n} = \hbar\omega\left(n+\frac{1}{2}\right)
\end{equation}
where ($n=0,1,2,\ldots$).
